It should be noted the current-day posture differs from the early Christian one: The traditional posture of prayer in medieval Europe is kneeling or supine with clasped hands, in antiquity more typically with raised hands. The early Christian prayer posture was standing, looking up to heaven, with outspread arms and bare head. This is the pre-Christian, pagan prayer posture (except for the bare head, which was prescribed for males in Corinthians 11:4, in Roman paganism, the head had to be covered in prayer).
As to the modern posture, closing the eyes during prayer, recitation or meditation is not unique to Christianity or the Abrahamic religions. It is well documented in older religions or traditions. The Pyramid Texts (2400 BCE) for example state in "charms and utterances":
429b. (with) face on the road, eye of (name of charmed person), look not at him.
In Buddhist practice (predating Christianity by about 600 years) it is prescribed to close the eyes entirely or almost completely (as illustrated by countless statues of the Buddha). The Buddhist canon probably has the most elaborate explanation for this action: closing the eyes help to turn the focus of the mind inward. This is called Vipassanā, a Pali word derived from the older prefix "vi-" meaning "special", and the verbal root "-passanā" meaning "seeing". It is often translated as "insight".
I was requested to provide some Buddhist canonical verses that illustrate closing the eyes (almost) completely was the common prescribed practice. As for the statues, you should keep in mind that in Hindu and Buddhist cultures, closed eyes symbolize looking inward towards the self.
In one of the oldest texts of the Tripitaka, the Vitakkasaṇṭhānasutta there is the direct instruction from the historical Buddha to his students:
As they do so, those bad thoughts are given up and come to an end.
Their mind becomes stilled internally; it settles, unifies, and
becomes immersed in samādhi. Suppose there was a person with good
eyesight, and some undesirable sights came into their range of vision.
They’d just close their eyes or look away. In the same way, a
mendicant … those bad thoughts are given up and come to an end …
(a mendicant (lit. "beggar") is a term for an early follower of the Buddha, who has given up all worldly possessions)
In the Sōtō Zen tradition of Dōgen, who brought Chan Buddhism to Japan, meditation is called zazen. In Japanese za means “sitting” and zen means “meditation”.
There are three important points of practice in sitting meditation. The first one is harmonizing the body, the second is harmonizing the breath, and the third is harmonizing the mind. Body, breath, and mind are the three most important points of practice in meditation. Posture is the key to practice. As for the eyes: you are to direct your vision about three feet in front of your body, and your eyes will naturally come to rest in position that is not completely closed (to prevent sleepiness).
In Shōbōgenzō: On Wanshi’s ‘Kindly Advice for Doing Seated Meditation’ Dōgen writes:
Do not esteem what is far off, and do not belittle what is far off;
just acquaint yourself with what is far off. Do not belittle what is
near at hand, and do not esteem what is near at hand; just acquaint
yourself with what is near at hand. Do not treat your eyes lightly,
and do not attach great importance to your eyes. And do not attach
great importance to your ears, and do not treat your ears lightly.
Just make your ears and your eyes sharp and clear...
The expression "make your ears and your eyes sharp and clear" means to not try to block out the senses, but just let them pass, without contemplating them or attaching words and meaning to them. This is why in most Zen schools, novices sit facing the wall with the eyes almost closed. Note that for advanced practitioners and walking meditation (practiced mostly in the Mahayana schools) the closed eyes prescription does not apply: once you have mastered turning the focus of the mind on itself, the senses will no longer distract your focus.
The eyes are then associated with clearly seeing the way things are, whereas the ears are associated with accurately understanding what things truly are. This is what Dōgen means further in the same text:
... Do not doubt your eyes and do not trust your ears
(allegorical for: do not doubt your direct experience of It and do not rely on your understanding of how someone else has described It.)
In the Tibetan Nyingma school, the instructions are a bit different, but with the same underlying goal:
keep the eyes open but not stare-open - don't strain to keep them
half-closed either, relax the muscles around the eyes, so called
soft-eyes, do not focus the eyes on anything in particular, allow them
go a little out of focus but don't force it, if your eyes wander from
object to object a bit, that's totally fine - as long as the objects
are not too engaging. You don't have to literally sit in front of the
wall but you want your scene to be rather generic.
As to the question on textual sources for Roman Pagan prayer posture and attire, Plutarch wrote the following on the matter:
“… they thus worshipped the Gods, either humbling themselves by concealing the head, or rather by pulling the toga over their ears as a precaution lest any ill-omened and baleful sound from without should reach them while they were praying...Or, as Castor states when he is trying to bring Roman customs into relation with Pythagorean doctrines: the Spirit within us entreats and supplicates the Gods without, and thus he symbolizes by the covering of the head the covering and concealment of the soul by the body - Plutarch, Roman Questions.”
As for the posture, this is often referred to as the orans position, which is Latin for “one who is praying.”
This was a common way of praying in the ancient world, not reserved to Christianity. In fact, most pagans prayed in the same way and pagan deities were visually represented standing or sitting in the orans position.